LESSON III.

DEFINITIONS, PRINCIPLES, AND RULES.

Art. 12. Arithmetic is the science of numbers, and the art of numerical computation.

A Number is a unit or a collection of units.

A Unit is one thing of any kind.

An Integer is a whole number.

Art. 13. There are three methods of expressing numbers:

1. By words; as, five, fifty, etc.

2. By letters, called the Roman method. (Art. 23.) 3. By figures, called the Arabic method.

Art. 14. Notation is the art of expressing numbers. by figures or letters.

Numeration is the art of reading numbers expressed by figures or letters.

The word Notation is commonly used to denote the Arabic method, which expresses numbers by figures.

Art. 15. In expressing numbers by figures, ten characters are used, viz.: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The first of these characters, 0, is called Naught, or Cipher. It denotes nothing, or the absence of number.

The other nine characters are called Significant Figures. They each express one or more units. They are also called Digits.

Art. 16. The successive figures which express a number, denote successive Orders of Units. These orders are numbered from the right; as, first, second, third, fourth, fifth, and so on.

A figure in units' place denotes units of the first order; in tens' place, units of the second order; in hundreds' place, units of the third order, and so on the term units being used to express ones of any order.

Art. 17. Ten units make one ten, ten tens make one hundred, ten hundreds make one thousand; and, generally, ten units of any order make one unit of the next higher order.

NOTE. The teacher can make this principle plain by means of the illustration given on page 9. It is easily shown that 10 ones or units equal 1 ten, and that 10 tens equal 1 hundred.

Art. 18. Figures have two values, called Simple and Local.

The Simple Value of a figure is its value when standing in units' place.

The Local Value of a figure is its value arising from the order in which it stands.

When 3, for example, stands alone, or in the first order, it denotes 3 units; when it stands in the second order, as in 34, it denotes 3 tens; when it stands in the third order, as in 354, it denotes 3 hundreds. Hence, the local value of figures increases from right to left in a tenfold ratio.

The local value of each of the successive figures which express a number, is called a Term. The terms of 325 are 3 hundredths, 2 tens, and 5 units.

Art. 19. The figures denoting the successive orders. of units, are divided into groups of three figures each, called Periods. The first or right-hand period is called Units; the second, Thousands; the third, Millions; the fourth, Billions; the fifth, Trillions; the sixth, Quadrillions; the seventh, Quintillions; etc.

I. A.-2.

Art. 20. The three orders of any period, counting from the right, denote, respectively, Units, Tens, and Hundreds, as shown in the table:

The several orders may be named more briefly by calling the first order of each period by the name of the period, and omitting the word "of" after tens and hundreds, thus:

Hundred-millions.

5th Period.

4th Period.

3d Period.

2d Period.

1st Period.

Art. 21. RULE for Notation.-Begin at the left, and write the figures of each period in their proper orders, filling all vacant orders and periods with ciphers.

Art. 22. RULE FOR NUMERATION -1. Begin at the right, and separate the number into periods of three figures each.

2. Begin at the left, and read each period containing one or more significant figures as if it stood alone, adding its name.

NOTE. The name of the units' period is usually omitted.

WRITTEN EXERCISES.

1. Write in words, 20080406.

SUGGESTION.-Separate the number into periods, thus: 20,080,406. Then write each period, thus: Twenty million eighty thousand four hundred and six.

2. Write in words, 50038456.
3. Write in words, 300607008.
4. Write in words, 40000300400.

SUGGESTION.-Omit the third period, since it contains no significant figures, thus: Forty billion three hundred thousand four hundred.

5. Write in words, 3450000067.
6. Read 3000080040; 10080603400.
7. Read 15000407030; 5075803004.

8. Read 400440300500; 130030003003.

9. Express in figures, twelve billion forty-six million and nine.

PROCESS. First, write 12, with a comma after it, to form the fourth or billions' period, thus: 12,; then write 46 in the next period, filling the vacant order with a cipher, thus: 12,046,; then, as there are no thousands, fill the next three orders with ciphers, thus: 12,046,000,; and, finally, write 9 in the units' period, filling the vacant orders with ciphers, thus: 12,046,000,009.

10. Express in figures, fifty million thirty-two thousand six hundred and forty.

11. Three hundred million nine thousand two hundred and six.

12. Forty-eight billion seventeen thousand sixty-four.

13. Five million five thousand and five.

and

14. One million one hundred thousand and ten. 15. Three trillion three hundred million three hundred and three.

16. Sixty-two million three hundred thousand and forty-nine.

17. Five hundred million five thousand.

18. Four hundred and six thousand five hundred and seven.

19. Two million ten thousand and eighty.

20. Ninety million seven thousand four hundred and ninety.

21. Four hundred million forty thousand four hundred and four.

22. Thirty billion seventy-five thousand.

23. Nine billion nine thousand and nine.

24. Fifty-four million eighty-seven thousand and eighty-six.

25. Two hundred and two thousand five hundred and eighty.

26. Fifty billion fifty million five hundred thousand and seven.

27. Seventeen billion seven hundred thousand three hundred and six.

28. Ninety million ten thousand and fifty-five.

LESSON IV.

ROMAN NOTATION.

Art. 23. In the Roman Notation, numbers are expressed by means of seven capital letters, viz.: I, V, X, L, C, D, M.